A Comparative Study of the Nonuniqueness

NASA" Technical Paper 2385

NASA-Tp-a385 19850007385

January 1985

A Comparative Study of the Nonuniqueness Problem of the Potential Equation M, D. Salas, A. Jameson, and R. E. Melnik

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NASA Technical Paper 2385 1985

A Comparative Study of the Nonuniqueness Problem of the Potential Equation M. D. Salas Langley Research Center Hampton, Virginia

A. Jameson Princeton University Princeton, New Jersey

R. E. Melnik Grumman Aerospace Corporation Bethpage, New York

NI_A

National Aeronautics and Space Administration

Scientific and Technical Information Branch

Summary The nonuniqueness problem occurring at transonic speeds with the conservative potential equation is investigated numerically. Evidence is given supporting tile thesis that the nonuniqueness problem is inherent in the potential differential equation rather than a consequence of the finite-difference approximation. Results are presented from an extensive comparative study between potential and Euler calculations for flow past two-dimensional airfoil profiles. This study indicates that the nonuniqueness problem is not an inviscid phenomenon, but results from the approximate treatment of shockA waves inherent inbound the conservative model. more restrictive on the limit potential of validity of the conservative

potential

model is suggested.

Introduction When the tangential stresses acting on a volume of fluid are small compared with the normal pressures exerted by the fluid, the fluid may be assumed to be inviscid. The mathematical expressions describing the conservation of mass, momentum, and energy of an inviscid flow are known as the Euler equations. In the absence of rotational effects, the inviscid flow can be described by a single conservation law, the conservation of mass, which can be expressed in terms of a function _, the velocity potential. The irrotational assumption is not always valid. Rotational effects can occur in an inviscid flow as a result of variations in free-stream stagnation conditions, because of the presence of shock waves, or because of satisfying boundary conditions, as for example in the case of a lifting delta wing with a sharp leading edge. For flow past two-dimensional airfoils immersed in a uniform upstream far field, rotational elfects are introduced at shock waves. Strictly speaking, within the inviscid assumption, these flows are governed by the Euler equations. However, if shock waves are sufficiently weak, the irrotational assumption should provide a reasonable approximation of the flow field. Shock waves, under this approximation, are replaced by sharp isentropic recompressions which approximate the true Rankine-Hugoniot shock up to second order in terms of the shock wave strength, Under the irrotational assumption, the conservation of mass equation can be written in quasi-linear or nonconservative form as (a2 - (I,_),I,_

- 2_%_y

+ (a 2 -

_y)_y_ 2 = 0]

a = a(¢_ + _5y2) or in conservation

_

-[- (P_y)y

fl = fl((i)2

+ dpy2)

(1)

a Cp

(2)

mass.being solved.

= 0 "_ j

son showed that numerical solutions of the full potential equation in conservative form for flow past an airfoil are not unique. In the same paper, a meticulous study was conducted which indicated that the anomaly is inherent in the partial differential equation and not a result of its discrete representation. Having reached this conclusion, the authors went on to conjecture the possibility of the anomaly being (1) an inviscid phenomenon, perhaps inherent to the mote exact Euler equations; and (2) perhaps a contributing factor in the occurrence of buffeting. Given the preeminent position today of the potential approximation in the transonic range for the design and analysis of airfoil profiles, as documented in reference 4, it is unnecessary to dwell on the importance of investigating the conjectures made in reference 3. Such an investigation is the purpose of this paper. The paper is divided into three main parts. First, a brief review of Steinhoff and Jameson's study and some new findings made in the course of this work are presented. Second, the Euler code described in reference 5 is introduced, and both improvements and the validation procedure used to determine its accuracy are described. Finally, results from a comparative study of the nonuniqueness problem for the potential and Euler models are presented and discussed. This paper provides results which are more comprehensive than the preliminary results reported in reference 6 and corrects some errors in an earlier version by the same title, reference 7.

Symbols

form as (P_x)x

where a is the speed of sound, p is the density, _ is the potential, and the subscripts denote partial derivatives. In the absence of shock waves, both forms of the equation are equivalent. When shock waves are present, use of the conservative form in conjunction with an "entropy" condition guarantees (see ref. 1) that the proper _ shock jump condition is satisfied. On the other hand, use of the nonconservative form leads to an ambiguous jump across the shock waves. With few exceptions, present-day state-of-the-art working transonic potential equation codes use the conservative form. For subcritical flows, Bers (ref. 2) has shown that the solution for flow past an airfoil section satisfying the Kutta condition is unique. In the transonic regime, little has been proven rigorously in terms of uniqueness. However, in a recent paper (ref. 3), Steinhoff and Jame-

local speed of sound pressure coefficient

1 By proper,

we mean

consistent

In this case, the model

with the mathematical requires

model

the conservation

of

c

airfoil chord

p

density

cd

section drag coefficient

7

ratio of airfoil thickness to airfoil chord

cl

section lift coefficient

_

velocity potential

Cm

section pitching-moment

¢

D

damping

reduced velocity potential, € = • - (aooMoo)(xcoso_ -I-ysin _)

X

transonic

operator

coefficient

defined by equa-

tion (6) Dx

damping

operator

defined by equa-

tion (7) Dy

damping tion (8)

similarity,

(1 - M2)/[T('7

+ 1)M2oo]2/3

Subscripts: operator

defined by equavalue downstream

F

far-field value

from shock wave

d

term defined by equation

h

cell area

i

cell index in X-direction

Jm_x

number of cells in Y-direction

j

cell index in Y-direction

L

wind-tunnel

o

value at which nonuniqueness

M

Mach number

s

value upstream

r

radial distance measured airfoil

c_

free-stream

S

similarity

t u

time component

v w

component of velocity in y-direction vector of conservative unknowns

X

computational

x

"O" mesh Cartesian coordinate

Y

(9)

d

half-height

from center of

of velocity in x-direction

coordinate

y

computational coordinate of "O" mesh Cartesian coordinate

a

angle of attack factor,

F

circulation

_/

ratio of specific heats

A €(2)

increment second-order coefficient

_(4)

fourth-order coefficient

0

polar angle measured

from shock wave

value

Review of the Potential Anomaly

entropy

compressibility

occurs

along ring of

normal to ring

(1 - M 2) 1/2

artificial damping

artificial damping counterclockwise

from trailing edge term defined by equation

While introducing lifting capability to the fullpotential, conservative, nonlifting, multigrid code described by Jameson in reference 8, Steinhoff and Jameson (ref. 3) discovered that three converged solutions could be obtained for a symmetric airfoil at zero angle of attack and a fixed free-stream Mach number. These solutions consisted of a zero-lift symmetric solution and two mirror-image asymmetric solutions with large absolute levels of lift. (See figs. 1 and 2 of ref. 3.) The latter two solutions could be obtained by perturbing the zero-lift solution, which was unstable. The multigrid capability of the code made it feasible to converge all three solutions to machine accuracY.2 This unquestionably demonstrated the multiplicity of solutions for the discrete system of equations. To determine if the anomaly was characteristic of the partial differential equation being modeled, the following tests were made in reference 3 with FLO 36: 1. The solutions were checked to determine if they had the expected far-field decay corresponding to a vortex and a doublet in a uniform stream (ref. 9). They did. In addition, the effect of the location of the outer boundary was investigated. It affected the lift level slightly, but the nonuniqueness problem persisted. 2. Originally, the anomaly was observed for an airfoil with nonzero trailing-edge included angle. To determine the effect of imposing the Kutta condition,

a Joukowski

2 Residuals

(13)

203 computer

were of the

profile order

was investigated. of 10 -12

for the eases reported

here.

The

in a CDC (_ CYBER

Joukowski profile, with its cusped trailing edge, requires a smooth matching of the pressures on the upper and lower surfaces. The numerical solutions showed the proper trailing-edge behavior, but again the anomaly persisted, 3. The convergence of these solutions with mesh refinement was investigated by using grids with 96 x 24, 128 x 32, 192 x 48, 256 x 64, and 384 x 192 cells in the X- and Y-directions, respectively. All solutions were found to change little with mesh refinement, 4. The effect of artificial viscosity was tested by using both first- and second-order accurate expressions for this term, with no effect on the anomaly, 5. Since all calculations were originally made with an "O" mesh defined by a conformal circle mapping, calculations were tried with a "C" mesh. The multiple solutions were also observed with the "C" mesh. 6. Similar results were obtained with the conservative finite-difference, potential code described in reference 10. In addition, during the course of the present investigation, we found the following results: 1. The multiple solutions also occurred with the conservative potential code described in reference 11 (TAIR). 2. A standard feature of these potential codes is the requirement that the reduced far-field potential Cr satisfy the leading term of the compressible vortex solution; namely, F tan_l[f_tan( /_ -- (1 - M2) 1/2

_ _ a)]

(3) (4)

where _ is the polar angle, and F is the value of the circulation that satisfied the Kutta condition. Because F evolved as part of the solution, it is reasonable to be suspicious of this boundary condition as a possible cause of the anomaly. To check this possibility, calculations were made with the "wind-tunnel" code described

exists at the differential equation level of the conservative potential formulation. Therefore, with no evidence to the contrary, we will proceed under that assumption. An interesting feature of the results presented in reference 3 was the occurrence of a "gap" in the lift curve. (See fig. 1.) This gap was explained in reference 3 in terms of a hysteresis effect. In the present investigation, it was determined that the gap comes about because the lift becomes a multivalued function of the angle of attack in this range. However, since the angle of attack remains a single-valued function of the lift, the problem can be made well posed by prescribing the lift and letting the numerical solution determine the angle of attack which satisfies the Kutta condition. (A similar technique was successfully applied by the principal author to another nonunique problem, ref. 15.) In this manner, the complete lift curve for an NACA 0012 airfoil has been evaluated at Moo -- 0.83 with FLO 36, a modified version of the code described in reference 3. The results are shown in figure 2. For the NACA 0012 airfoil, the results presented in figure 2 show the anomaly occurring in the neighborhood of the zero angle of attack. It should not be inferred from this that the anomaly occurs only at this Mach number and narrow range of angles of attack. At any given Mach number in the transonic regime, a range of angles of attack can be found where the nonuniqueness occurs. For example, for the NACA 0012 airfoil at Moo = 0.79, it appears at around a -- 1°; at Moo = 0.60, it appears at around a = 9°. It has been speculated that the anomaly is a result of an interaction between an upper-surface shock and a lower-surface shock. However, the anomaly has been observed in a number of cases in which only a single shock (either on the lower or upper surface) is present. A case in point is the flow past an RAE 2822 profile at Moo = 0.75 and a = 1°. The lift curve for these conditions is shown in figure 3. The single shock wave occurs on the upper surface, as shown in figure 4.

in reference 12. In this code, the far-field boundary condition corresponds to no flow through the tunnel walls; that is,

Nonuniqueness is not new to potential theory. An incompressible flow past a lifting airfoil with a sharp trailing edge has a one-parameter family of solutions. In

= 0 (5) y=+L Oy Essentially "free-air" solutions are generated with L -50c. The multiple solutions were also found with this code. 3. The same anomalous behavior was found with the conservative, Cartesian coordinate code described in reference 13. 4. No multiple solutions were found with the nonconservative code described in reference 14 (FLO 12). Although the above tests are not conclusive, they make a strong case for the argument that the problem

the Kutta condition. At supercritical speeds, solutions this case, the relevant physical solution is singled out by with "expansion" shocks are mathematically valid solutions of the governing differential equation. Expansion shocks are ruled out in favor of the physically relevant compression shock by invoking the Second Law of Thermodynamics. In fact, these constraints are now considered integral parts of a numerical algorithm. Why not resolve the present dilemma by imposing some new constraint? If we consider figure 2 again, at zero angle of attack, the symmetric solution corresponding to zero lift seems to be the solution that we would want to single out as relevant with the new constraint. However,

0¢r

consider what happens if the angle of attack is increased by a small amount. Which of these solutions is physically relevant? The two with the largest absolute levels of lift can be ruled out on the grounds that they require a discontinuous

coordinate in the normal direction. is given by Dw = Dxw+

The damping

term

Dyw

(6)

behavior as a goes to 0°. However, the

Dxw

= di+l/2j

- di-1/2,j

(7)

remaining solution does not seem physically relevant either, since it predicts a negative lift-curve slope. The nonuniqueness problem we are facing is unique in that none of the solutions available seem to be physically relevant[ Indeed, if a new constraint is found, its role will

Dyw

= di,j+l/2

- di,j-1/2

(8)

not be to single of thesolution. three solutions available, but to out findone a new

presently

The question now is, Is this anomaly a problem of the conservative potential approximation in the transonic range or a problem of the inviscid flow? To investigate this question, we propose to do a systematic search with an Euler flow code. However, before proceeding to this step, it is necessary to establish the validity of the numerical results of the Euler code. This is the next step in our investigation.

Improvements and Validation of FLO 52-S The finite-volume, four-step Runge-Kutta code described in reference 5 (FLO 52) was chosen for this study because of its fast convergence rate. The code solves the time-dependent continuity, x-momentum, ymomentum, and energy equations in conservative form. To prevent typical central-difference, odd-even oscillations andto capture shock waves, a blended fourth-and second-order explicit artificial damping term is added to the equations. To accelerate the rate of convergence, each mesh cell is advanced at its own local time step, At, and the governing equations are modified by an enthalpy damping term. The calculations were done with a nearly orthogonal "O" mesh with an inverse radial transformation to cluster mesh points near the airfoil, A detailed account of the discretization, damping, and acceleration technique is given in reference 5. In its original version, the code consistently underpredicted the lift coefficient from potential calculations by as much as 10 percent for lifting subcritical flows. In addition, isomach plots revealed a disturbing behavior near the airfoil surface. As the isomach lines approached the surface, they turned abruptly. This turning indicated an incorrect "boundary-layerlike" behavior. This was also evident in flow-field plots of entropy and total pressure losses. The cause of the problem was found to be associated with the evaluation of the damping terms near the airfoil surface and in the far field. As described in reference 5, the damping term is made up of contributions from the two coordinate directions X and Y with associated indexes i and j, respectively. Let X be the coordinate along a ring of the "O" mesh and Y be the

4

where w is the vector of conservative typical d term is defined by

di,j+l/2

-

unknowns,

and a

hi,j+1/2 re(2) .,, [ /j+1/2 _wi,j+l/2 At -6_4)+1/2(Awi,j+3/2 1

-2 Awl,j+1/2 (9)

+ n?zi,j-1/2)J

The symbol h represents

the cell area, 6(2) and 6(4) are

coefficients, defined in reference 5, which depend on the second derivative of pressure, and Awl,j+1/2

=

ZOi,jq-1

--

ZVi,j

(10)

The evaluation of the d terms presents some problems near the airfoil surface_ j < 2, and near the last ring of the "O" mesh, j > Jma× - 1, because of the lack of information near these cells. In the original code, Aw terms inside the airfoil and outside the last ring of the "O" mesh were constructed with a third-order extrapolation from inside the flow field. The "boundary-layerlike" behavior was eliminated by the following procedure. At a ghost point j = 0 inside the airfoil, the pressure, density, and total enthalpy were obtained by linear extrapolation from the flow field. The velocity components at the cell center immediately next to the airfoil surface were decomposed into components normal and tangent to the surface. These were then reflected to obtain components at the ghost cell. With this information, it was possible to evaluate Awe,l/2 = w_,l - w_,o The missing term Aw_,-1/2 was evaluated by equating it to AWl,l/2. Although this procedure eliminated the "boundary-layerlike" behavior, it had little effect on the low-lift level being predicted: Surprisingly, the problem with the lift was corrected by modifying the evaluation of the far-field damping terms. The finitevolume integration is performed only up to Jmax - 1; values of w at Jmax are obtained by satisfying the far-field boundary conditions, which use no damping. Therefore, to evaluate the damping terms at Jmax - 1, only additional information for the term Awi,gmax.l/2 is required. When this term was approximated by Awi,j.,,x+l/2 = Awi,j,,,,x-1/2 the error in the lift level was reduced to about 1 percent.

The far-field boundary condition evaluation in the original code used the method of Rudy and Strikwerda (ref. 16), which introduces an additional free parameter into the calculation. This method was replaced by the following procedure, which proved more robust and made a slight improvement in the convergence rate. At a subsonic point, a frame of reference perpendicular and tangential to the last ring of the "O" mesh is constructed. In the plane made by the perpendicular to the ring of the "O" mesh and the time axis, the Riemann "invariant" that propagates along the characteristic coming into the computational region is prescribed from known free-stream values. The Riemann "invariant" propagated by the outgoing characteristic is extrapolated from computed values. From these two quantities, the speed of sound and the velocity component normal to the ring of the "O" mesh are determined. If the point corresponds to an inflow point, the entropy and velocity components tangent to the ring of the "O" mesh are prescribed from free-stream values. If the point corresponds to an outflow point, the entropy and velocity components tangent to the ring of the "O" mesh are extrapolated from computed values. From these four quantities, the conservative unknown vector w is evaluated. For supersonic inflow, all quantities are prescribed, but for supersonic outflow, all quantities are extrapolated from computed values, The code was also modified to include, as an option, the effect of a far-field vortex. To this end, the circulation is approximated from the lift calculated by the integration of the surface pressure. Then, the free-stream Cartesian velocity components modified to account for the vortex are defined as

sisted of a second-order accurate integration of the normal pressure gradient at the surface of the airfoil to evaluate the surface pressure. As discussed in reference 5, because of the finite-volume formulation and the fact that the fluxes convected across the airfoil surface are zero, only the pressure needs to be evaluated at the surface. In this connection, it should be stressed that the evaluation of ghost points, discussed in relation to the damping terms, is not related to the evaluation of the surface boundary conditions. For reference purposes, we will designate the modified Euler code as FLO 52-S. A large number of lifting subcritical cases were calculated with this code by using a 120 × 34 mesh and with the potential code FLO 36 by using a 192 x 32 mesh. Results are presented for one of the "most difficult" subcritical lifting cases computed--an NACA 0012 airfoil at Moo -- 0.30 and a = i0 °. A partial view of the mesh corresponding to 120 × 34 points is shown in figure 5. Figure 6 shows that second-order accuracy in the lift and drag coefficients is attainable with mesh refinement. Figure 7 shows a typical convergence behavior. The residuals plotted in figure 7(a) are the root-mean-square deviations of Ap/At, Apu/At, and Apv/At evaluated over the entire field. Figure 7(b) shows the convergence of lift, drag, and moment coefficients, each evaluated from the integration of the surface pressure. A comparison between FLO 36 and FLO 52-S for the surface pressure distribution, isomach contours, and streamline trajectories is shown in figures 8 to 10. Practically no difference can be observed if the results of the two calculations are overlaid; we emphasize the results presented correspond to a case that showed the most disagreement with a potential calculation.

too = uoo- _sinO _oo = Vco +_cos0

At supercritical speeds, the potential CalcUlations are only approximate solutions to the inviscid flow. Typically, these calculations have a stronger shock wave, which appears further aft than in Euler calcu-

(11) (12)

where flF = 27rr[1 - M 2 sin2(0 -a)] 1 F = -_caooMooct

lations.

Two typical results

at Moo = 0.82 and 0.86

(13)

(both at zero angle of attack) are shown in figures 11 and 12. A check on how well the shock wave is captured

(14)

is shown in figure 13, where the calculated shock jump is compared with the exact jump. Taking into consideration the difficulty in reading the results because of the shock-wave smearing and the Zierep singularity, the agreement is quite good. For comparison, the results of the nonconservative potential code, FLO 12, are also included.

and r, 0 are polar coordinates in the physical plane with the origin at the center of the airfoil. The free-stream thermodynamic variables are recomputed by using the magnitude of the modified free-stream velocity given by equations (11) and (12), the steady-state Bernoulli equation, and the known value of entropy at the point in question. In general, the far-field vortex had very little effect on the calculated results. This is probably the result of the location of the last ring of the "O" mesh, which in these calculations corresponded to about 100 chords from the airfoil, The last major modification to the Euler code con-

Nouliulqllelless Study The lift curve shown in figure 2 is repeated in figure 14, along with the results from the Euler calculation (FLO 52-S) and the nonconservative potential code FLO 12. As can be seen from this figure, the Euler results do not show any anomaly at this particular

5

Mach number. Can it be that the Euler solution will be anomalous at some other Mach number? In order to study this, consider the angle made by the lift curve with the abscissa at zero lift. If this angle exceeds 90 °, then we are into the anomalous region, since a positive angle of attack produces negative lift. Figure 15 shows the behavior of this angle as a function of free-stream Mach number for conservative potential, nonconservative potential, and Euler calculations. The figure shows that the conservative potential solution becomes progressively worse as the free-stream Mach number increases and eventually crosses over into the multiple solution region at about Moo = 0.82. Both Euler and nonconservative potential solutions remain well-behaved throughout the Mach number range. The

Moo = 0.820, a = 0°, and flow past an NACA 0006 airfoil at Moo -- 0.879, a = 0° both correspond to the same value of the transonic similarity parameter X. This value corresponds to conditions on the verge of nonuniqueness. It is reasonable to expect that if for one of these flow fields the potential assumption ceases to be valid, then it should cease to be valid for the other flow field as well. The computed surface Mach numbers for these two cases are illustrated in figures 17 and 18. It is evident from figure 18 that the thinner airfoil does not violate the classical criterion. This should be expected, since relation (15) is not scaled according to similarity rules. The problem can be overcome if we do not allow the similarity-scaled entropy

anomaly, therefore, is indicative of a breakdown of the conservative potential formulation, not of inviscid

S _ (M2 - 1)3 T2

theory, per se. In figure 16, the plot of figure 15 is repeated in terms of transonic similarity parameters (ref. 17). In addition to the results obtained for an NACA 0012 airfoil, results for an NACA 0006 airfoil are also shown in this figure. It is interesting to see that the conservative potential results appear to collapse into a single curve in the similarity plane. This is another indication that the numerical code is predicting the behavior expected of the partial differential equation. It furthermore indicates that the nonuniqueness problem will occur even for very weak shock waves if the airfoil thickness T is made sufficiently small. The fact that the Euler results do not collapse into a single curve in the similarity plane is not unexpected, since the similarity law is only valid for the small-disturbance potential equation. The rapid drop in Ocl/Oa shown in figure 16 for X < 0.8 implies a drop in lift, which has been explained as an inviscid phenomenon associated with the inefficient production of lift that occurs when the upper surface shock wave moves to the trailing edge (ref. 17 and ref. 18, p. 656). Classically, potential theory is accepted as a good approximation to the rotational inviscid flow in the transonic range if the shock is weak. The shock strength is measured by M 2 - 1, where Ms is the Mach number upstream of the shock. The rationale is based on the fact that the entropy produced at the shock is proportional to the cube of the shock strength. Thus, the potential approximation is accepted if

to be greater than the value it reaches when the nonuniqueness occurs. For example, for an NACA 0012 airfoil near a -- 0°, nonuniqueness is observed at a value of Ms of approximately 1.3; with this information, we can compute from relation (16) a value of S which we will designate So. For any other affine profile of thickness r, we must require that the Mach number at the shock obey the constraint

(M 2 -1)3